- #36

jbriggs444

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If I follow the semantics correctly, we start with the well-accepted: ##\log_b x = \frac{\log x}{\log b}##

Equally well accepted is: ##\ln x = \log_e x = \frac{\log x}{\log e}##

I see two possible ways to systematically extend this notation scheme to supply a meaning for ##\ln_b x##.

Suppose that we define the meaning of

In particular, ##\ln_2 x = \frac{\ln x}{\ln 2} = \frac {\frac{\log x}{\log e}} {\frac{\log 2}{\log e}} = \frac{\log x}{\log 2} = \log_2 x##

So this possible definition would mean that there is no such thing as a log with two bases. Only logs with one base.

Suppose, alternately, that we define the meaning of

In particular, ##\ln_2 x = \frac{\ln x}{\log 2} = \frac {\frac{\log x}{\log e}} {\log 2} = \frac {\log x}{\log 2e} = \log_{2e}x##

Apparently this was the selected meaning. Ludicrous.

Equally well accepted is: ##\ln x = \log_e x = \frac{\log x}{\log e}##

I see two possible ways to systematically extend this notation scheme to supply a meaning for ##\ln_b x##.

Suppose that we define the meaning of

**any**subcripted function ##f_b(x)## to be ##\frac{f(x)}{f(b)}##In particular, ##\ln_2 x = \frac{\ln x}{\ln 2} = \frac {\frac{\log x}{\log e}} {\frac{\log 2}{\log e}} = \frac{\log x}{\log 2} = \log_2 x##

So this possible definition would mean that there is no such thing as a log with two bases. Only logs with one base.

**Any subscript renders the original base irrelevant.**Suppose, alternately, that we define the meaning of

**any**subscripted function ##f_b(x)## to be ##\frac{f(x)}{\log b}##In particular, ##\ln_2 x = \frac{\ln x}{\log 2} = \frac {\frac{\log x}{\log e}} {\log 2} = \frac {\log x}{\log 2e} = \log_{2e}x##

Apparently this was the selected meaning. Ludicrous.

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